Risk Difference vs Odds Ratio in Logistic Regression: A Quick Stata User Guide (with Stata Cheats)

TL;DRLogit shows log-odds, logistic shows odds ratios. Risk ≠ Odds; Risk Difference (RD) ≠ Odds Ratio (OR). Model choice flows from outcome type and clinical question. Use margins for predicted risk and RD; use glm, link(identity) to model RD directly.

1. One Model, Two Faces

Stata command	What it really models	Default output	Typical use-case
logit	log-odds (β)	β coefficients	building models, plotting
logistic	same equation as logit	exp(β) (OR)	reporting interpretable effect sizes

Think of them as identical twins wearing different name-tags.

* Same fit, different dress code
logit    outcome exposure covariates        // β (log-odds)
logistic outcome exposure covariates        // OR

2. Risk, Odds & Log-Odds – The Mental Map

Probability (Risk)   →  Odds (p / 1-p)   →  Log-Odds (log(odds))
  0 ⇢ 1                 0 ⇢ ∞               –∞ ⇢ +∞  (straight line)

Risk Difference (RD) = linear gap in probability (easy to grasp).
Odds Ratio (OR) = multiplicative change in odds (can exaggerate when outcome common).
Log-Odds = what the regression actually handles; linear in X, so estimation is easy.

3. Quick Model Picker

Outcome	Preferred model	Stata verbs
Binary (0/1)	Logistic regression(OR or RD)	logistic, logit, glm
Continuous	Linear regression	regress
Counts	Poisson / NB	poisson, nbreg
Time-to-event	Survival	stcox, streg

Rule-of-thumb: If clinicians ask, “How many times higher?” → OR/RR. If they care, “How much absolute risk change?” → RD.

4. Getting Risk & Risk Difference from a Logistic Fit

* Fit logistic model (any flavour)
logit died i.exposure age sex bp

* A. Individual predicted risk
predict p_death, pr           // probability for each person

* B. Group risks + RD in one shot
margins exposure              // mean risk (p) by exposure
margins exposure, dydx(exposure)  // RD_exposed – RD_unexposed

* Output: risk_unexp, risk_exp, RD, 95% CI

Direct RD modelling (optional)

glm died i.exposure age sex bp, family(binomial) link(identity) vce(robust)
* β_exposure is the RD (difference in probability)

Tip: Always tack on vce(robust) with the identity link—the variance formula is quirky.

5. Mini-Example

Imagine a cohort where treatment cut 28-day mortality from 12 % to 8 %.

. logit death i.treat age sex

. margins treat
--------------------------------------------------------------------------
        |            Delta-method
        |     Margin   Std. Err.    z    P>|z|     [95% Conf. Interval]
-------------+------------------------------------------------------------
  treat |
     0  |   .1200      .0078     15.40   0.000     .1048     .1352
     1  |   .0800      .0061     13.06   0.000     .0681     .0918
--------------------------------------------------------------------------

. margins treat, dydx(treat)
--------------------------------------------------------------------
        |            Delta-method
        |         dy/dx       Std. Err.   z    P>|z|  [95% Conf. Interval]
-------------+----------------------------------------------------------
  treat |  (1 vs 0)   -.0400    .0099    -4.04   0.000   -.0594   -.0206
--------------------------------------------------------------------

RD = –4 % (absolute benefit).
Convert to Number-Needed-to-Treat: NNT ≈ 1/0.04 = 25.
For comparison, logistic would show an OR ≈ 0.64 for the same effect.

6. When RD Beats OR (and vice-versa)

Scenario	Recommend
Outcome frequency <10 %	OR or RR fine (they converge)
Clinicians want absolute impact	RD (NNT pops right out)
Meta-analysis of diverse settings	Log-OR (statistically handy)
Policy / public-health framing	RD, Risk Ratio, or ARR

Message: Match the metric to the decision.

7. A Decision Flow (20-second check)

Binary outcome?
    └── Yes ──► Is absolute risk change what matters?
                    ├─ Yes: model RD (glm identity or margins)
                    └─ No : report OR (logistic) or RR (modified Poisson)
Continuous outcome?
    └── Use linear regression (regress)
Count outcome?
    └── Poisson/NB (poisson, nbreg)
Time-to-event?
    └── Cox / parametric survival

8. Common Pitfalls (Rapid Fire)

OR ≠ RR when outcome common – avoid surprise.
glm, link(identity) needs robust SE.
Perfect prediction? Combine sparse categories or use firthlogit.
For adjusted risk or RD, always follow with margins – it standardises across covariates.
Report CI and sample size alongside any metric; NNT alone hides precision.

9. Copy-Paste Stata Cheat Sheet

* Basic logistic with OR output
logistic death i.treat age sex comorbid, or

* Predicted risk by treat group
margins treat

* Risk difference
margins treat, dydx(treat)

* Direct RD model
glm death i.treat age sex, family(binomial) link(identity) vce(robust)

* Modified Poisson for adjusted RR
glm death i.treat age sex, family(poisson) link(log) vce(robust) eform

*--- 2×2 cohort table: RISK RATIO & RISK DIFFERENCE --------------------------
cs outcome exposure, risk          // RR, RD, AR%  (+ exact CI if add option exact)
cs outcome exposure, risk exact    // exact (Fisher) CI

*--- 2×2 case-control table: ODDS RATIO --------------------------------------
cc case exposure, woolf            // OR (+ Woolf CI; add exact for Fisher)
cc case exposure, exact            // exact (Cornfield/Fisher) CI

*--- Extra bells -------------------------------------------------------------
cs outcome exposure, by(strata)    // stratified RR (M-H pooled)
cc case exposure, by(strata)       // stratified OR (M-H pooled)

cs (cohort study) estimates Risk Ratio, Risk Difference, Attributable Risk% from a 2×2 table. cc (case-control) estimates Odds Ratio from a 2×2 table.

10. Wrap-Up

Same equation, different lens: choose logit for model development, logistic for quick ORs.
Translate coefficients into risk (with margins) whenever the audience thinks in probabilities.
Risk Difference speaks the language of clinicians and guidelines; get it either via margins ... dydx() or a GLM with identity link.
Let the clinical question drive the statistic, not the other way around.

Happy modelling – and may your odds be ever in your favour (or your risks differ in the right direction).